On Integer Programming, Discrepancy, and Convolution

Integer programs with a constant number of constraints are solvable in pseudo-polynomial time. We give a new algorithm with a better pseudo-polynomial running time than previous results. Moreover, we establish a strong connection to the problem (min, +)-convolution. (min, +)-convolution has a trivial quadratic time algorithm and it has been conjectured that this cannot be improved significantly. We show that further improvements to our pseudo-polynomial algorithm for any fixed number of constraints are equivalent to improvements for (min, +)-convolution. This is a strong evidence that our algorithm's running time is the best possible. We also present a faster specialized algorithm for testing feasibility of an integer program with few constraints and for this we also give a tight lower bound, which is based on the SETH.Comment: A preliminary version appeared in the proceedings of ITCS 201

Local Search Breaks 1.75 for Graph Balancing

Graph Balancing is the problem of orienting the edges of a weighted multigraph so as to minimize the maximum weighted in-degree. Since the introduction of the problem the best algorithm known achieves an approximation ratio of 1.75 and it is based on rounding a linear program with this exact integrality gap. It is also known that there is no (1.5 - epsilon)-approximation algorithm, unless P=NP. Can we do better than 1.75? We prove that a different LP formulation, the configuration LP, has a strictly smaller integrality gap. Graph Balancing was the last one in a group of related problems from literature, for which it was open whether the configuration LP is stronger than previous, simple LP relaxations. We base our proof on a local search approach that has been applied successfully to the more general Restricted Assignment problem, which in turn is a prominent special case of makespan minimization on unrelated machines. With a number of technical novelties we are able to obtain a bound of 1.749 for the case of Graph Balancing. It is not clear whether the local search algorithm we present terminates in polynomial time, which means that the bound is non-constructive. However, it is a strong evidence that a better approximation algorithm is possible using the configuration LP and it allows the optimum to be estimated within a factor better than 1.75. A particularly interesting aspect of our techniques is the way we handle small edges in the local search. We manage to exploit the configuration constraints enforced on small edges in the LP. This may be of interest to other problems such as Restricted Assignment as well

On Integer Programming and Convolution

Integer programs with a constant number of constraints are solvable in pseudo-polynomial time. We give a new algorithm with a better pseudo-polynomial running time than previous results. Moreover, we establish a strong connection to the problem (min, +)-convolution. (min, +)-convolution has a trivial quadratic time algorithm and it has been conjectured that this cannot be improved significantly. We show that further improvements to our pseudo-polynomial algorithm for any fixed number of constraints are equivalent to improvements for (min, +)-convolution. This is a strong evidence that our algorithm\u27s running time is the best possible. We also present a faster specialized algorithm for testing feasibility of an integer program with few constraints and for this we also give a tight lower bound, which is based on the SETH

Compact LP Relaxations for Allocation Problems

We consider the restricted versions of Scheduling on Unrelated Machines and the Santa Claus problem. In these problems we are given a set of jobs and a set of machines. Every job j has a size p_j and a set of allowed machines Gamma(j), i.e., it can only be assigned to those machines. In the first problem, the objective is to minimize the maximum load among all machines; in the latter problem it is to maximize the minimum load. For these problems, the strongest LP relaxation known is the configuration LP. The configuration LP has an exponential number of variables and it cannot be solved exactly unless P = NP. Our main result is a new LP relaxation for these problems. This LP has only O(n^3) variables and constraints. It is a further relaxation of the configuration LP, but it obeys the best bounds known for its integrality gap (11/6 and 4). For the configuration LP these bounds were obtained using two local search algorithm. These algorithms, however, differ significantly in presentation. In this paper, we give a meta algorithm based on the local search ideas. With an instantiation for each objective function, we prove the bounds for the new compact LP relaxation (in particular, for the configuration LP). This way, we bring out many analogies between the two proofs, which were not apparent before

Algorithms for Integer Programming and Allocation

The first part of the thesis contains pseudo-polynomial algorithms for integer linear programs (ILP). When certain parameters of an ILP are fixed, that is, they are treated as constants in the running time, it is possible to obtain algorithms with a running time that is pseudo-polynomial in the entries of the ILP’s matrix. We present a tight pseudo-polynomial running time for ILPs with a constant number of constraints. Furthermore, we study an extension of this model to MILPs (linear programs that contain both fractional and integer variables). Then we move to n-fold ILPs, a class of ILPs with block structured matrices. We present the first algorithm for n-folds, which is near-linear in the dimensions of the ILP. The second part is about scheduling in non-identical machine models, more precisely, restricted allocation problems. Here a set of jobs has to be allocated to a set of machines. However, every job has a subset of machines and may only be assigned to a machine from this subset. We consider the objectives of minimizing the makespan or maximizing the minimum load. We study the integrality gap of a particularly strong linear programming relaxation, the configuration LP, for variations of this problem. The integrality gap can be seen as a measure of strength of an LP relaxation. A local search technique can be used to bound this value. However, the proofs are generally non-constructive, i.e., they do not give an efficient approximation algorithm right away. We derive better upper bounds on the integrality gap of the problems Restricted Assignment, Restricted Santa Claus, and Graph Balancing. Furthermore, we give the first (constructive) quasi-polynomial time approximation algorithm for Restricted Assignment with an approximation ratio strictly less than 2.Der erste Teil der Thesis umfasst pseudopolynomielle Algorithmen für ganzzahlige lineare Programme (ILP). Wenn bestimmte Parameter eines ILPs fixiert sind, d.h. sie werden in der Laufzeit als Konstanten betrachtet, dann ist es möglich Algorithmen zu entwerfen, deren Laufzeit pseudopolynomiell in dem größten absoluten Wert eines Eintrags der Matrix des ILPs ist. Ein Ergebnis, das wir präsentieren, ist eine scharfe Schranke für die pseudopolynomielle Laufzeit, die nötig ist um ein ILP mit konstant vielen Bedingungen zu lösen. Danach befassen wir uns mit n-fold ILPs, eine Klasse von ILPs, deren matrix eine Blockstruktur besitzt. Wir geben den ersten Algorithmus für n-folds an, dessen Laufzeit gleichzeitig nahezu linear in der Dimension des ILPs ist. Der zweite Teil handelt von nicht-identischen (heterogenen) Maschinen Modellen, genauer gesagt restricted allocation problems. Hier soll eine Menge von Jobs auf eine Menge von Maschinen verteilt werden. Jeder Job darf aber nur auf bestimmte Maschinen zugewiesen werden. Wir betrachten als Zielfunktionen sowohl die Minimierung des Makespans als auch die Maximierung der minimalen Last einer Maschine. Wir untersuchen den integrality gap einer besonders starken LP Relaxierung, dem Konfigurations LP, für Variationen dieses Problems. Der integrality gap kann als Maß für die Stärke einer LP Relaxierung gesehen werden. Über ein Argument mittels einer lokalen Suche wird dieser Wert beschränkt. Jedoch sind die Beweise typischerweise nicht konstruktiv, d.h. sie implizieren nicht direkt effiziente Approximationsalgorithmen. Wir beweisen neue obere Schranken an den integrality gap für die Probleme Restricted Assignment, Restricted Santa Claus und Graph Balancing. Desweiteren präsentieren wir den ersten (konstruktiven) Quasipolynomialzeit Approximationsalgorithmus für das Restricted Assignment Problem mit Approximationsrate echt kleiner als 2

Near-Linear Time Algorithm for n-fold ILPs via Color Coding

We study an important case of ILPs max {c^Tx | Ax = b, l <= x <= u, x in Z^{n t}} with n * t variables and lower and upper bounds l, u in Z^{nt}. In n-fold ILPs non-zero entries only appear in the first r rows of the matrix A and in small blocks of size s x t along the diagonal underneath. Despite this restriction many optimization problems can be expressed in this form. It is known that n-fold ILPs can be solved in FPT time regarding the parameters s, r, and Delta, where Delta is the greatest absolute value of an entry in A. The state-of-the-art technique is a local search algorithm that subsequently moves in an improving direction. Both, the number of iterations and the search for such an improving direction take time Omega(n), leading to a quadratic running time in n. We introduce a technique based on Color Coding, which allows us to compute these improving directions in logarithmic time after a single initialization step. This leads to the first algorithm for n-fold ILPs with a running time that is near-linear in the number nt of variables, namely (rs Delta)^{O(r^2s + s^2)} L^2 * nt log^{O(1)}(nt), where L is the encoding length of the largest integer in the input. In contrast to the algorithms in recent literature, we do not need to solve the LP relaxation in order to handle unbounded variables. Instead, we give a structural lemma to introduce appropriate bounds. If, on the other hand, we are given such an LP solution, the running time can be decreased by a factor of L

The Submodular Santa Claus Problem in the Restricted Assignment Case

The submodular Santa Claus problem was introduced in a seminal work by Goemans, Harvey, Iwata, and Mirrokni (SODA\u2709) as an application of their structural result. In the mentioned problem n unsplittable resources have to be assigned to m players, each with a monotone submodular utility function f_i. The goal is to maximize min_i f_i(S_i) where S?,...,S_m is a partition of the resources. The result by Goemans et al. implies a polynomial time O(n^{1/2 +?})-approximation algorithm. Since then progress on this problem was limited to the linear case, that is, all f_i are linear functions. In particular, a line of research has shown that there is a polynomial time constant approximation algorithm for linear valuation functions in the restricted assignment case. This is the special case where each player is given a set of desired resources ?_i and the individual valuation functions are defined as f_i(S) = f(S ? ?_i) for a global linear function f. This can also be interpreted as maximizing min_i f(S_i) with additional assignment restrictions, i.e., resources can only be assigned to certain players. In this paper we make comparable progress for the submodular variant: If f is a monotone submodular function, we can in polynomial time compute an O(log log(n))-approximate solution